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I'm going to suggest this might be Cestus Dei by John Maddox Roberts.
The entire settled galaxy is divided among the various major religions, that meet as a body called the "Unified Faiths" to mediate disputes and resolve problems.
The opening paragraph is:
Archbishop Hilarion occupied his rightful seat in the Great Hall of United Faiths with the ...

This sounds remarkably similar to the article written by Harvard law professor Roger Fisher in the Bulletin of Atomic Scientists in 1981. The article is available here. The article includes:
Put that needed code number in a little capsule, and then implant that capsule right next to the heart of a volunteer. The volunteer would carry with him a big, heavy ...

I believe this is Marvel Adventures: The Avengers #26
It's the bottom of the ninth, the Avengers are down by two, two strikes down and bases loaded. The stakes for the Cosmic Pennant have never been higher. How will Galactus call this one? Okay... we admit. We have no idea what's going on in this story. Kirk drew the cover, and now Parker has to come up ...

Could this be The Resurrected Man by Sean Williams?
The protagonist is a one-time private eye named Jonah whose former lover and colleague Marylin, now a cop, he must work with to track down a serial killer known as the Twinmaker. Since all of the victims resemble Marylin, he is maybe a suspect, but because he was found damaged, and was in suspension for 3 ...

This sounds like Casshern Sins (2008). The elements that fit are
Aired on Adult Swim/Cartoon Network
The world is mostly a wasteland. Humanity is nearly extinct
Robots are afraid of dying via rust
Robots fare little better, fearing death as much as humans do: the poisonous environment quickly causes their mechanical bodies to rust and corrode, forcing ...

It could also be Soul Frame Lazenca (1997).
In the beginning of the 22nd Century, nuclear war has devastated the Earth and turned it into a wasteland. The few survivors gather and raise the city-state Setos, and while progressing a revival plan for mankind, come into contact with the Atman lifeforms sleeping in the planet's mantle.

The book is indeed part of John Cleve's Spaceways series (John Cleve was a pen name of Andrew J. Offutt). It is Of Alien Bondage, which is the first book in the series.
The book starts just after the protagonist Janja has been captured. Her husband has already been killed by this point:
Janja wasn't the strangest piece of cargo he'd carried, but she was ...

The Mote in God's Eye (1974), by Larry Niven and Jerry Pournelle
After an encounter with an alien slower-than-light probe ends violently, a human interstellar empire sends an expedition to the aliens' solar system. The human name for the aliens' home star is "The Mote", so the aliens come to be known as "Moties".
Early contact with civilization at the ...

This sounds a little like Brent Week's Night Angel series. The information given is a closest match for the opening of book 2 Shadow's Edge. Though the publication date is way off. As the Night Angel series was first published starting in 2008.
Shadow's Edge opens in Cenaria, a medieval city state that has been invaded by the Godking of Khalidor a ...

I believe this to be one of various books from L.J. Smith's "Night of the Solstice" series. If I'm correct, this actually involves three sisters and one brother.
From the author's homepage (http://www.ljanesmith.net/author/booklist/197-the-night-of-the-solsticeheart-of-valor), note this blurb, in which the sisters are training (one of them under Morgana to ...

Sounds like The Ship of Ishtar by A. Merritt
The archaeologist hero, Kenton, receives a mysterious ancient
Babylonian artifact, which he discovers contains an incredibly
detailed model of a ship. A dizzy spell casts Kenton onto the deck of
the ship, which becomes a full-sized vessel sailing an eternal sea. At
one end is Sharane the assistant ...

Larry Niven, The Integral Trees (1984) and The Smoke Ring (1987)
Humans have colonized a free-fall ("zero gravity") biosphere orbiting a neutron star. Some of the humans live in pseudogravity at the ends of enormous trees larger than skyscrapers, while others inhabit free-floating structures. In the absence of strong planetbound gravity, these colonists ...

This looks like Triplet by Tim Zahn.
The main characters are a researcher (Danae) and an escort (Ravagin) who are entering through gateways into alternate versions of a world.
Danae is nominally studying the words but actually studying the impact of travelling between worlds (and being in the alternate worlds) on the escorts.
The bodyguard (set on her by ...

That's "The Mick of Time" by Spider Robinson, the last story in the collection Callahan's Secret. It was first published in the May 1986 issue of Analog, although it was by then edited by Stanley Schmidt, not John W Campbell. It's a short story not a novel, and it's not really hard SF, but it otherwise fits your description accurately.
The paragraph in ...

This is Interceptor Force (1999).
In the near future, the government has created an elite team of
soldiers specially trained in the event of a hostile alien encounter.
A multi-national force of the world's best, they are known as the
Interceptor Force. A covert early warning satellite network tracks an
object crash landing in a town in the Pacific ...

After posting my last answer, I found a second match with the same search that might better match, albeit from a later timeframe. Quest, by Thomas Stellmach, was released in 1996 and according to one review does involve him falling through to other worlds.
It is claymation with very good stop motion and interesting backgrounds of a character made out of ...

Could this be the 1977 Le château de sable (Sand Castle)?
o Hoedeman’s Oscar winning animated short is a charming story about creation, diversity and teamwork. A man made of sand creates a bizarre assortment of sand-based characters to help him build a beautiful sand castle they can all call home. The stop-frame animation may appear simplistic but it is ...

Personal Jesus by Jennifer Pelland. I found it in the anthology the page is based on just today.
Here's a summary:
"Personal Jesus" is an electronic watchdog that everyone must wear, advising proper religious behavior and applying electrical shocks when wearers misbehave. A vignette written as a promotional pamphlet.

Circa 1958, I read a short story in a Sci-Fi anthology
Perhaps "Quietus", a short story by Ross Rocklynne, first published in Astounding Science-Fiction, September 1940, which is available at the Internet Archive. If you read it in an anthology around 1958, it must have been one edition or another of Famous Science-Fiction Stories: Adventures in Time and ...

Could it be Felix the Cat? The theme song for the 1950s TV animation went
Felix the cat
The wonderful, wonderful cat
Whenever he gets in a fix
He reaches into his bag of tricks
and that just about describes every episode of the cartoon. The character was from an earlier print cartoon, but the bag of tricks gimmick was introduced by the TV version.

Some parts of your description sound like Mistborn by Brandon Sanderson:
The series primarily takes place in a region called the Final Empire on a world called Scadrial, where the sun and sky are red, vegetation is brown, and the ground is constantly being covered under black volcanic ashfalls. Every night the land becomes covered in an unnatural mist or ...

The Demons at Rainbow Bridge (1989) by Jack L. Chalker?
From this Goodreads review:
In this series, the galaxy is split into three mutually antagonistic power blocs, the Exchange, the religious nutters of the Mizlaplanian Empire, and the evil dog-eat-dog empire of the Mycohlians. Humanity went out into the stars and found itself just another alien race ...

Doomwatch: Winter Angel, 1999. IMDb lists it as a movie, though; but it is the continuation of a 70s series of the same name.
Neil Tannahill a University lecturer is asked by the former head of Doomwatch (now very old and dying) to investigate the strange happenings at a remote nuclear facility. Dangerous nuclear waste from around the world is being ...

This is the (made-for-TV) movie Doomwatch: Winter Angel (1999), based on the Cult 1970s TV series of the same name.
Neil arrives on the scene and finds that something is going very wrong - and that the facilities staff are desperate for his help. The research staff - whilst experimenting with particle physics - have created a tiny black hole - which they ...

This is “To Sin Against Systems” by Garry R. Osgood, published on Isaac Asimov’s Science Fiction Magazine, 4th issue.
Got it all, the rich guy, the immortal who takes over when he dies, the space station I didn't remember, the promises to work for the better good by the rich man, the way he dies...
The story can be read online at the Internet Archive here.

While it wasn't my intent to corner the market on answers to questions about sexually powered interstellar drives, I just read a story that is very likely to actually be the answer that was sought.
Namely, "Thrust" by Alan Dean Foster.
In this story, humanity's first interstellar ship is expected to reach its destination in just over 16 years. But there's ...

In Marion Zimmer Bradley's Darkover series, Leonie and Lorrill Hastur are twins in (one of) the highest ranking family on the planet. She is the stronger telepath and undergoes rigorous training which in the end brings her to the most powerful position in the "Towers", where psychic powers are trained and used. Lorill goes on to be the Regent for Prince ...

That's Huntik, an American-Italian cartoon aired between 2009 and 2012.
From left to right, meet Sophie, Lok, Dante and Zhalia (who's not yet a woman but can be perceived as such).
I remember something that he could spawn down monsters (I remember one monster which was like a floating eye something like that).
The four main characters are "Seekers"; ...

Since “Colossus” is a partial match/partial mismatch, there is another story that is a partial match/partial mismatch in different ways. And mfrank66 has not gotten back as promised to say whether Colossus was the sought answer.
Question refers to “…a science fiction book about earth were [sic] two supercomputers were buried deep down under the surface.
One ...

I think the book with the handprint on the cover is called The Power.
In The Power the world is a recognisable place: there's a rich Nigerian kid who lounges around the family pool; a foster girl whose religious parents hide their true nature; a local American politician; a tough London girl from a tricky family. But something vital has changed, causing ...

That is Flight into Space edited by Don Wollheim and published in 1950 by Frederick Fell.
It's subtitled "Great Science-Fiction Stories of Interplanetary Travel" and just as you describe, has a story for the Sun, all nine planets, the Moon and the asteroids. The Sun creatures (in Sunward by Stanton A. Coblentz) are as you said, but the Plutonians (The Rape ...

This is most likely “And Then There Were (N-One)” by Sarah Pinsker
The viewpoint character is an insurance claims investigator
There is a murder
All characters involved in the murder (victims, investigator, suspects) are alternate timeline versions of the main character
The only difference is that in appeared in Uncanny, issue 15, March/April 2017. You can ...

The Universe Between, a 1965 novel by Alan E. Nourse, which was also the answer to this old question. It was based on the short story "High Threshold" in Astounding Science Fiction, March 1951 and the novelette "The Universe Between" in Astounding Science Fiction, September 1951, which are available at the Internet Archive here and here.
ISFDB synopsis:
...

I was just looking for this too - was reading it a few years ago, and really liked it but then kind of stopped and forgot where it was. I came across your post and was disappointed that it didn't lead to the answer. But I think I found it: https://www.gunnerkrigg.com/

Is it about 2 young girls who find a Ouija board and end up contacting a soldier from long ago? This soldier is actually the ghost of one of the girls soulmates. Then the story goes into her perspective as an adult and she learns that their love has been cursed and in every life they end up torn apart. If so, I totally remember this story and have also been ...

Could this be Bad Magic by Stephan Zielinski (2004)?
Magic is real. Strange forces and uncanny creatures abound. Cosmic
evil is loose in the world--especially around San Diego--along with
transcendent evil, primordial evil, simmering evil, objective evil,
and specific evil.
The main character created a cloak of invisibility by pulling threads of ...

Could this be Hidden by Megg Jensen (part of the Dragonlands series of books)?
The mystery enshrouding Hutton’s Bridge is as impenetrable as the fog
that descended at its borders eighty years ago. Each year, three
villagers enter the mist searching for answers. No one ever returns.
Then a dragon falls from the sky to the town square, dead—the ...

I'm about ready to bet a pig's butt to a C-note that it is
"The Language Clarifier" by Paul J. Nahin
... and aiming to find out. Unlike some answered stories, this one is being difficult to find and read again today to make sure. (Or add those demanded and coveted block-quotes from 30 years back in memory.)
This one was indeed in Omni.
Searching cross-...

Passage for Piano by Frank Herbert.
The main protagonist is Margaret Hatchell:
Had some cosmic crystal gazer suggested to Margaret Hatchell that she would try to smuggle a concert grand piano onto the colony spaceship, she would have been shocked. Here she was at home in her kitchen on a hot summer afternoon, worried about how to squeeze ounces into her ...

That's Guy Gavriel Kay's Fionavar Tapestry
From Wikipedia (with my emphasis):
The Five [travellers]:
Drawn by magic from our world and thus strangers to Fionavar, each of them finds a new role and a new destiny during their adventures in Fionavar.
-Kevin Laine (Liadon) – Witty, bright, outgoing. Fair of hair and of spirit. The act of love has ...

That figure sounds like Matt Mason and this link confirms there was a tie-in book with moon worms:
The writing is iffy at best but wildly, and I mean WILDLY imaginative:
moon rabbits, moon worms, "nothing" beings--and all the great Major
Matt Mason vehicles.
It appears to have been a Whitman Big Little Book, and is listed on a collector's info page ...